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7.3 Constant Relative Radial Speed

Let ξ(t) be the projection of the vector PR(t) − PT(t − D(t)) along the propagation direction ik (oriented from PT(t − D(t)) to PR(t)), that is

(7.77)

The delay D(t) experienced by the wavefront transmitted at time t − D(t) in PT(t − D(t)) and received at time t in PR(t) depends on the distance ||PR(t) − PT(t − D(t))||. Accordingly with (7.11) we have

As observed in Section 7.1.3, |ξ(t)| is the time-varying distance responsible of the Doppler effect, viz, the time-varying delay D(t), and should not be confused with

(7.79)

Let us assume that the relative radial speed between PT(t − D(t)) and PR(t) is constant within the observation interval. Thus, the projection of the vector PR(t) − PT(t − D(t)) along the propagation direction ik is a linear function of t. That is,